Properties

Label 3450.2.a.t.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{27} -4.00000 q^{28} -4.00000 q^{29} +1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} +6.00000 q^{41} -4.00000 q^{42} -2.00000 q^{43} -2.00000 q^{44} -1.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} -6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000 q^{58} +12.0000 q^{59} -14.0000 q^{61} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} -2.00000 q^{67} -2.00000 q^{68} -1.00000 q^{69} -2.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -10.0000 q^{74} +8.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -8.00000 q^{83} -4.00000 q^{84} -2.00000 q^{86} -4.00000 q^{87} -2.00000 q^{88} -8.00000 q^{89} -1.00000 q^{92} -12.0000 q^{94} +1.00000 q^{96} +9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −4.00000 −0.617213
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −2.00000 −0.242536
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) −4.00000 −0.428845
\(88\) −2.00000 −0.213201
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 9.00000 0.909137
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −2.00000 −0.198030
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) −4.00000 −0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −14.0000 −1.26750
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) −2.00000 −0.167836
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) 9.00000 0.742307
\(148\) −10.0000 −0.821995
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −4.00000 −0.308607
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 12.0000 0.901975
\(178\) −8.00000 −0.599625
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) −12.0000 −0.875190
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −2.00000 −0.142134
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) −12.0000 −0.844317
\(203\) 16.0000 1.12298
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) −2.00000 −0.137038
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) −10.0000 −0.671156
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) −4.00000 −0.262613
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 8.00000 0.519656
\(238\) 8.00000 0.518563
\(239\) 22.0000 1.42306 0.711531 0.702655i \(-0.248002\pi\)
0.711531 + 0.702655i \(0.248002\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) −4.00000 −0.251976
\(253\) 2.00000 0.125739
\(254\) −6.00000 −0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −2.00000 −0.124515
\(259\) 40.0000 2.48548
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 0 0
\(263\) 32.0000 1.97320 0.986602 0.163144i \(-0.0521635\pi\)
0.986602 + 0.163144i \(0.0521635\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) −2.00000 −0.122169
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) 0 0
\(281\) 32.0000 1.90896 0.954480 0.298275i \(-0.0964112\pi\)
0.954480 + 0.298275i \(0.0964112\pi\)
\(282\) −12.0000 −0.714590
\(283\) 10.0000 0.594438 0.297219 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −6.00000 −0.351123
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) −2.00000 −0.116052
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) −12.0000 −0.690522
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 8.00000 0.455842
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) −20.0000 −1.13047 −0.565233 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) −6.00000 −0.336463
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 14.0000 0.774202
\(328\) 6.00000 0.331295
\(329\) 48.0000 2.64633
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −8.00000 −0.439057
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 4.00000 0.217894 0.108947 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) −13.0000 −0.707107
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −4.00000 −0.214423
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) 8.00000 0.423405
\(358\) 4.00000 0.211407
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −6.00000 −0.315353
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) −2.00000 −0.101666
\(388\) 0 0
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) −32.0000 −1.60603 −0.803017 0.595956i \(-0.796773\pi\)
−0.803017 + 0.595956i \(0.796773\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 16.0000 0.794067
\(407\) 20.0000 0.991363
\(408\) −2.00000 −0.0990148
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) −4.00000 −0.197066
\(413\) −48.0000 −2.36193
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 4.00000 0.194717
\(423\) −12.0000 −0.583460
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) 56.0000 2.71003
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) −24.0000 −1.15337 −0.576683 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) −14.0000 −0.662177
\(448\) −4.00000 −0.188982
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) −6.00000 −0.282216
\(453\) −12.0000 −0.563809
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 24.0000 1.12267 0.561336 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) −26.0000 −1.21490
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 8.00000 0.372194
\(463\) −18.0000 −0.836531 −0.418265 0.908325i \(-0.637362\pi\)
−0.418265 + 0.908325i \(0.637362\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 12.0000 0.552345
\(473\) 4.00000 0.183920
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) −6.00000 −0.274721
\(478\) 22.0000 1.00626
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −14.0000 −0.637683
\(483\) 4.00000 0.182006
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) −14.0000 −0.633750
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 6.00000 0.270501
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) −8.00000 −0.358489
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) −13.0000 −0.577350
\(508\) −6.00000 −0.266207
\(509\) 28.0000 1.24108 0.620539 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) 24.0000 1.05552
\(518\) 40.0000 1.75750
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) −4.00000 −0.175075
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) −8.00000 −0.346194
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 4.00000 0.172613
\(538\) 24.0000 1.03471
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 8.00000 0.343629
\(543\) −6.00000 −0.257485
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 2.00000 0.0854358
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) −32.0000 −1.36078
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) −26.0000 −1.10166 −0.550828 0.834619i \(-0.685688\pi\)
−0.550828 + 0.834619i \(0.685688\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 32.0000 1.34984
\(563\) −8.00000 −0.337160 −0.168580 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) −4.00000 −0.167984
\(568\) −2.00000 −0.0839181
\(569\) −12.0000 −0.503066 −0.251533 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −13.0000 −0.540729
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 32.0000 1.32758
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) −10.0000 −0.410997
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 8.00000 0.326056
\(603\) −2.00000 −0.0814463
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) 16.0000 0.648353
\(610\) 0 0
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) −4.00000 −0.160904
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −2.00000 −0.0801927
\(623\) 32.0000 1.28205
\(624\) 0 0
\(625\) 0 0
\(626\) −20.0000 −0.799361
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 8.00000 0.318223
\(633\) 4.00000 0.158986
\(634\) 10.0000 0.397151
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 8.00000 0.316723
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) 12.0000 0.473602
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 1.00000 0.0392837
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −6.00000 −0.234082
\(658\) 48.0000 1.87123
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) 6.00000 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) 28.0000 1.08093
\(672\) −4.00000 −0.154303
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) 4.00000 0.154074
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) −26.0000 −0.991962
\(688\) −2.00000 −0.0762493
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000 0.228086
\(693\) 8.00000 0.303895
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) −4.00000 −0.151620
\(697\) −12.0000 −0.454532
\(698\) −30.0000 −1.13552
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) 48.0000 1.80523
\(708\) 12.0000 0.450988
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −8.00000 −0.299813
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 22.0000 0.821605
\(718\) 0 0
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −19.0000 −0.707107
\(723\) −14.0000 −0.520666
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) −14.0000 −0.517455
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 4.00000 0.147342
\(738\) 6.00000 0.220863
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) −8.00000 −0.292705
\(748\) 4.00000 0.146254
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) −12.0000 −0.437595
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 4.00000 0.145287
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) −6.00000 −0.217357
\(763\) −56.0000 −2.02734
\(764\) 0 0
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 14.0000 0.503871
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 0 0
\(777\) 40.0000 1.43499
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 2.00000 0.0715199
\(783\) −4.00000 −0.142948
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) 34.0000 1.21197 0.605985 0.795476i \(-0.292779\pi\)
0.605985 + 0.795476i \(0.292779\pi\)
\(788\) 6.00000 0.213741
\(789\) 32.0000 1.13923
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) −2.00000 −0.0710669
\(793\) 0 0
\(794\) −32.0000 −1.13564
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −50.0000 −1.77109 −0.885545 0.464553i \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) −12.0000 −0.423735
\(803\) 12.0000 0.423471
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) 24.0000 0.844840
\(808\) −12.0000 −0.422159
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) 16.0000 0.561490
\(813\) 8.00000 0.280572
\(814\) 20.0000 0.701000
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) −16.0000 −0.558404 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(822\) 2.00000 0.0697580
\(823\) 10.0000 0.348578 0.174289 0.984695i \(-0.444237\pi\)
0.174289 + 0.984695i \(0.444237\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) −8.00000 −0.277517
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −26.0000 −0.898155
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −6.00000 −0.206774
\(843\) 32.0000 1.10214
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 28.0000 0.962091
\(848\) −6.00000 −0.206041
\(849\) 10.0000 0.343199
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) −2.00000 −0.0685189
\(853\) −56.0000 −1.91740 −0.958702 0.284413i \(-0.908201\pi\)
−0.958702 + 0.284413i \(0.908201\pi\)
\(854\) 56.0000 1.91628
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 24.0000 0.817443
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −24.0000 −0.815553
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 14.0000 0.474100
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 56.0000 1.89099 0.945493 0.325643i \(-0.105581\pi\)
0.945493 + 0.325643i \(0.105581\pi\)
\(878\) 20.0000 0.674967
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 40.0000 1.34763 0.673817 0.738898i \(-0.264654\pi\)
0.673817 + 0.738898i \(0.264654\pi\)
\(882\) 9.00000 0.303046
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −10.0000 −0.335578
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −2.00000 −0.0669650
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) −34.0000 −1.13459
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −12.0000 −0.399556
\(903\) 8.00000 0.266223
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −12.0000 −0.398673
\(907\) −42.0000 −1.39459 −0.697294 0.716786i \(-0.745613\pi\)
−0.697294 + 0.716786i \(0.745613\pi\)
\(908\) −12.0000 −0.398234
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 24.0000 0.793849
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 20.0000 0.658665
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) −18.0000 −0.591517
\(927\) −4.00000 −0.131377
\(928\) −4.00000 −0.131306
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) −2.00000 −0.0654771
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 0 0
\(937\) −48.0000 −1.56809 −0.784046 0.620703i \(-0.786847\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 8.00000 0.261209
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) 22.0000 0.716799
\(943\) −6.00000 −0.195387
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) 8.00000 0.259281
\(953\) −14.0000 −0.453504 −0.226752 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 22.0000 0.711531
\(957\) 8.00000 0.258603
\(958\) −40.0000 −1.29234
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 4.00000 0.128698
\(967\) −58.0000 −1.86515 −0.932577 0.360971i \(-0.882445\pi\)
−0.932577 + 0.360971i \(0.882445\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) 50.0000 1.60458 0.802288 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) 1.00000 0.0320750
\(973\) −80.0000 −2.56468
\(974\) −18.0000 −0.576757
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 16.0000 0.511624
\(979\) 16.0000 0.511362
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) −24.0000 −0.765871
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 8.00000 0.254772
\(987\) 48.0000 1.52786
\(988\) 0 0
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 0 0
\(993\) −28.0000 −0.888553
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) −40.0000 −1.26681 −0.633406 0.773819i \(-0.718344\pi\)
−0.633406 + 0.773819i \(0.718344\pi\)
\(998\) −28.0000 −0.886325
\(999\) −10.0000 −0.316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3450.2.a.t.1.1 1
5.2 odd 4 3450.2.d.n.2899.2 2
5.3 odd 4 3450.2.d.n.2899.1 2
5.4 even 2 690.2.a.b.1.1 1
15.14 odd 2 2070.2.a.s.1.1 1
20.19 odd 2 5520.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.b.1.1 1 5.4 even 2
2070.2.a.s.1.1 1 15.14 odd 2
3450.2.a.t.1.1 1 1.1 even 1 trivial
3450.2.d.n.2899.1 2 5.3 odd 4
3450.2.d.n.2899.2 2 5.2 odd 4
5520.2.a.r.1.1 1 20.19 odd 2